On (𝑛 + ½)-Engel groups

Abstract

Abstract Let n be a positive integer. We say that a group G is an ( n + 1 2 ) {(n+\frac{1}{2})} -Engel group if it satisfies the law [ x , y n , x ] = 1 {[x,{}_{n}y,x]=1} . The variety of ( n + 1 2 ) {(n+\frac{1}{2})} -Engel groups lies between the varieties of n-Engel groups and ( n + 1 ) {(n+1)} -Engel groups. In this paper, we study these groups, and in particular, we prove that all ( 4 + 1 2 ) {(4+\frac{1}{2})} -Engel { 2 , 3 } {\{2,3\}} -groups are locally nilpotent. We also show that if G is a ( 4 + 1 2 ) {(4+\frac{1}{2})} -Engel p-group, where p ≥ 5 {p\geq 5} is a prime, then G p {G^{p}} is locally nilpotent.